Is food poisoning dangerous to an unborn baby in the first trimester?

Depends on mom. Baby is dependent on mom for everything. Anything that puts mom at risk puts the baby at risk. The most severe risk with food poisoning (non-bacterial) is dehydration from vomiting and diarrhea. As long as mom can maintain hydration and a small amount of nutrition, most food poisonous are limited and should resolve quickly without harm to the fetus.
Yes. The toxins from food poisoning will not traverse the placenta so the baby will not get sick directly. However, food poisoning can result in dehydration which can be problematic.